Digital Signal Processing Reference
In-Depth Information
TABLE 5.1
Fourier Transform Properties
Operation
Time Function
Fourier Transform
Linearity
Time shift
af
1
(t) + bf
2
(t)
aF
1
(v) + bF
2
(v)
F(v)e
-jvt
0
f(t - t
0
)
1
ƒ
a
ƒ
v
a
Time scaling
f(at)
F
¢
≤
1
ƒ a ƒ
v
a
¢
≤
e
- jvt
0
/a
Time transformation
f(at - t
0
)
F
Duality
F(t)
2pf(-v)
f(t)e
jv
0
t
Frequency shift
Convolution
F(v-v
0
)
f
1
(t)*f
2
(t)
F
1
(v)F
2
(v)
1
2p
F
1
(v)*F
2
(v)
f
1
(t)f
2
(t)
d
n
[f(t)]
dt
n
(jv)
n
F(v)
Differentiation
d
n
[F(v)]
dv
n
(-jt)
n
f(t)
t
1
jv
F(v) + pF(0)d(v)
Integration
f(t)dt
L
- q
then
[af
1
(t) + bf
2
(t)]
Î
f
"
[aF
1
(v) + bF
2
(v)],
(5.10)
where
a
and
b
are constants. In words, the principle of superposition applies to the
Fourier transform.
The linearity property of the Fourier transform
EXAMPLE 5.2
We can make use of the property of linearity to find the Fourier transforms of some types of
waveforms. For example, consider
f(t) = B cos
v
0
t.
Using Euler's relation,
e
ja
+
e
-ja
2
cos
a =
,
we can rewrite the expression for
f(t)
as
B
2
[e
jv
0
t
+ e
-jv
0
t
] =
B
2
e
jv
0
t
+
B
2
e
-jv
0
t
.
f(t) =
